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Search: a093112 -id:a093112
Displaying 1-6 of 6 results found.
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A091515
Numbers k such that (2^k - 1)^2 - 2 = 4^k - 2^(k+1) - 1 is prime.
+10
17
2, 3, 4, 6, 7, 10, 12, 15, 18, 19, 21, 25, 27, 55, 129, 132, 159, 171, 175, 315, 324, 358, 393, 435, 786, 1459, 1707, 2923, 6462, 14289, 39012, 51637, 100224, 108127, 110953, 175749, 185580, 226749, 248949, 253987, 520363, 653490, 688042, 695631
(list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
LINKS
Table of n, a(n) for n=1..44.
Steven Harvey, Carol and Kynea Primes
M. Rodenkirch, Carol and Kynea Prime Search
Eric Weisstein's World of Mathematics, Near-Square Prime
Eric Weisstein's World of Mathematics, Integer Sequence Primes
MATHEMATICA
lst={}; Do[p=(2^n-1)^2-2; If[PrimeQ[p], AppendTo[lst, n]], {n, 7!}]; lst (* Vladimir Joseph Stephan Orlovsky, Jan 27 2009 *)
PROG
(PARI) is(n)=ispseudoprime((2^n - 1)^2 - 2) \\ Charles R Greathouse IV, Feb 19 2016
CROSSREFS
Cf. A093112, A091516.
KEYWORD
nonn,hard
AUTHOR
Eric W. Weisstein, Jan 17 2004
EXTENSIONS
More terms from Pab Ter (pabrlos(AT)yahoo.com), May 25 2004
a(36)=175749 from Cletus Emmanuel (cemmanu(AT)yahoo.com), Oct 08 2004
a(37)=185580 from Cletus Emmanuel (cemmanu(AT)yahoo.com), Nov 03 2004
Edited by Ray Chandler, Nov 15 2004
a(38)=226749 from Steven Harvey, Jan 11 2005 and subsequently confirmed as next term
a(39) from Eric W. Weisstein, Mar 31 2006
a(40) = 253987 from Cletus Emmanuel (cemmanu(AT)yahoo.com), May 03 2007
a(41) = 520363 from Eric W. Weisstein, Jun 08 2016 (computed by Mark Rodenkirch)
a(42) = 653490 from Eric W. Weisstein, Jun 15 2016 (computed by Mark Rodenkirch)
a(43) = 688042 from Mark Rodenkirch, Jul 05 2016
a(44) = 695631 from Mark Rodenkirch, Jul 16 2016
STATUS
approved
A093069
a(n) = (2^n + 1)^2 - 2.
+10
9
7, 23, 79, 287, 1087, 4223, 16639, 66047, 263167, 1050623, 4198399, 16785407, 67125247, 268468223, 1073807359, 4295098367, 17180131327, 68720001023, 274878955519, 1099513724927, 4398050705407, 17592194433023, 70368760954879, 281475010265087, 1125899973951487
(list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
COMMENTS
Cletus Emmanuel calls these "Kynea numbers".
Difference between the smallest digitally balanced number with 2n+4 binary digits and the largest digitally balanced number with 2n+2 binary digits (see A031443): 7 = 9-2 = 1001-10, 23 = 35-12 = 100011-1100, 79 = 135-56 = 10000111-111000 etc. - Juri-Stepan Gerasimov, Jun 01 2011
LINKS
Michael De Vlieger, Table of n, a(n) for n = 1..1660
Amelia Carolina Sparavigna, Binary Operators of the Groupoids of  OEIS A093112 and A093069 Numbers(Carol and Kynea Numbers), Department of Applied Science and Technology, Politecnico di Torino (Italy, 2019).
Amelia Carolina Sparavigna, Some Groupoids and their Representations by Means of Integer Sequences, International Journal of Sciences (2019) Vol. 8, No. 10.
Eric Weisstein's World of Mathematics, Near-Square Prime
Index entries for linear recurrences with constant coefficients, signature (7,-14,8).
FORMULA
a(n) = 4^n+2^(n+1)-1.
G.f.: -x*(7-26*x+16*x^2) / ( (x-1)*(2*x-1)*(4*x-1) ). - R. J. Mathar, Jun 01 2011
a(n) = A092431(n+2) - A020522(n+1). - R. J. Mathar, Jun 01 2011
E.g.f.: -exp(x) + 2*exp(2*x) + exp(4*x) - 2. - Stefano Spezia, Dec 09 2019
EXAMPLE
G.f. = 7*x + 23*x^2 + 79*x^3 + 287*x^4 + 1087*x^5 + 4223*x^6 + 16639*x^7 + ...
MAPLE
A093069:=n->(2^n+1)^2-2: seq(A093069(n), n=1..30);
MATHEMATICA
a[ n_] := If[ n < 1, 0, 4^n + 2^(n + 1) - 1]; (* Michael Somos, Jul 08 2014 *)
CoefficientList[Series[(7 - 26*x + 16*x^2)/((1 - x)*(2*x - 1)*(4*x - 1)), {x, 0, 30}], x] (* Wesley Ivan Hurt, Jul 08 2014 *)
PROG
(PARI) vector(100, n, (2^n+1)^2-2) \\ Colin Barker, Jul 08 2014
(PARI) Vec(-(16*x^2-26*x+7)/((x-1)*(2*x-1)*(4*x-1)) + O(x^100)) \\ Colin Barker, Jul 08 2014
(Magma) [(2^n+1)^2-2 : n in [1..30]]; // Wesley Ivan Hurt, Jul 08 2014
CROSSREFS
Cf. A091514 (primes of the form (2^n + 1)^2 - 2).
Cf. A244663.
KEYWORD
nonn,easy
AUTHOR
Eric W. Weisstein, Mar 17 2004
EXTENSIONS
More terms from Colin Barker, Jul 08 2014
STATUS
approved
A098879
a(n) = (2^n - 1)^5 - 2.
+10
1
-2, -1, 241, 16805, 759373, 28629149, 992436541, 33038369405, 1078203909373, 34842114263549, 1120413075641341, 35940921946155005, 1151514816750309373, 36870975646169341949, 1180231376725002502141, 37773167607267111108605, 1208833588708967444709373
(list; graph; refs; listen; history; text; internal format)
OFFSET
0,1
COMMENTS
5th-power analog of what for exponent 2 is A093112 (2^n-1)^2 - 2 = 4^n - 2^{n+1} - 1 and exponent 3 is A098878 (2^n - 1)^3 - 2. Primes include a(n) for n = 0, 2, 5, 6. These are "near-5th-power prime." Semiprimes include a(n) for n = 3, 8, 9, 10, 13, 15, 21, 29, 33, 40. - Jonathan Vos Post, May 03 2006
LINKS
Harvey P. Dale, Table of n, a(n) for n = 0..664
Eric Weisstein's World of Mathematics, Near-Square Prime.
Index entries for linear recurrences with constant coefficients, signature (63, -1302, 11160, -41664, 64512, -32768).
FORMULA
G.f.: (-2+125*x-2300*x^2+22640*x^3-57728*x^4+66560*x^5)/((-1+x)(-1+32*x)(-1+16*x)(-1+8*x)(-1+4*x)(-1+2*x)). - R. J. Mathar, Nov 14 2007
EXAMPLE
If n=2, (2^2 - 1)^5 - 2 = 241 (a prime).
MATHEMATICA
(2^Range[0, 20]-1)^5-2 (* or *) LinearRecurrence[{63, -1302, 11160, -41664, 64512, -32768}, {-2, -1, 241, 16805, 759373, 28629149}, 20] (* Harvey P. Dale, Nov 03 2016 *)
PROG
(PARI) a(n)=(2^n-1)^5-2 \\ Charles R Greathouse IV, Feb 19 2016
CROSSREFS
Cf. A091516, A091515, A098878, A091514.
KEYWORD
easy,sign
AUTHOR
Parthasarathy Nambi, Oct 13 2004
EXTENSIONS
More terms from Jonathan Vos Post, May 03 2006
Edited by N. J. A. Sloane, Sep 30 2007
STATUS
approved
A244845
Binary representation of 4^n - 2^(n+1) - 1.
+10
1
111, 101111, 11011111, 1110111111, 111101111111, 11111011111111, 1111110111111111, 111111101111111111, 11111111011111111111, 1111111110111111111111, 111111111101111111111111, 11111111111011111111111111, 1111111111110111111111111111
(list; graph; refs; listen; history; text; internal format)
OFFSET
2,1
LINKS
Colin Barker, Table of n, a(n) for n = 2..500
Wikipedia, Carol number
Index entries for linear recurrences with constant coefficients, signature (111,-1110,1000).
FORMULA
a(n) = 111*a(n-1)-1110*a(n-2)+1000*a(n-3).
a(n) = (-1-9*10^(1+n)+100^n)/9.
G.f.: x^2*(89000*x^2-88790*x-111) / ((x-1)*(10*x-1)*(100*x-1)).
EXAMPLE
a(3) is 101111 because A093112(3) = 47 which is 101111 in base 2.
MATHEMATICA
Table[FromDigits[IntegerDigits[4^n-2^(n+1)-1, 2]], {n, 2, 15}] (* Harvey P. Dale, Oct 03 2016 *)
PROG
(PARI) vector(100, n, (100^(n+1)-9*10^(2+n)-1)/9)
(PARI) Vec(x^2*(89000*x^2-88790*x-111)/((x-1)*(10*x-1)*(100*x-1)) + O(x^100))
(PARI) a(n) = subst(Pol(binary(4^n-2^(n+1)-1)), x, 10); \\ Michel Marcus, Jul 08 2014
CROSSREFS
Cf. A093112.
KEYWORD
nonn,base,easy
AUTHOR
Colin Barker, Jul 07 2014
STATUS
approved
A331858
a(n) = (2^p-1)*(2^(p-1))*((2^p-1)^2-2), where p is the n-th prime.
+10
1
42, 1316, 475664, 131080256, 8783210218496, 2250975213522944, 147570574898545885184, 37778715690312487141376, 2475879193127080196116054016, 41538374636164863806350357434466304, 10633823951424046514111736193740701696, 178405961584350762488394070192754827810832384
(list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
COMMENTS
Integers a(1), a(2), a(4), a(8) corresponding to p = 2, 3, 7, 19 are also terms of A331805. - Bernard Schott, Feb 04 2020
LINKS
Table of n, a(n) for n=1..12.
FORMULA
a(n) = A060286(n)*A093112(prime(n)). - M. F. Hasler, Jan 31 2020
MATHEMATICA
f[p_] := (2^p-1)*(2^(p-1))*((2^p-1)^2-2); f @ Prime @ Range[12] (* Amiram Eldar, Jan 29 2020 *)
PROG
(PARI) [(2^p-1)*((2^p-1)^2-2)<<(p-1) | p<-primes(12)] \\ or: a(n, p=prime(n))={...}. - M. F. Hasler, Jan 29 2020
CROSSREFS
Cf. A000040 (primes), A000396 (perfect numbers), A093112 ((2^n-1)^2-2), A060286 (2^(p-1)*(2^p-1)), A331805.
KEYWORD
nonn,easy
AUTHOR
G. L. Honaker, Jr., Jan 29 2020
STATUS
approved
A118558
a(n) = (2^n-1)^4 - 2.
+10
0
-2, -1, 79, 2399, 50623, 923519, 15752959, 260144639, 4228250623, 68184176639, 1095222947839, 17557851463679, 281200199450623, 4501401006735359, 72040003462430719, 1152780773560811519, 18445618199572250623, 295138898083176775679, 4722294425687923097599
(list; graph; refs; listen; history; text; internal format)
OFFSET
0,1
LINKS
Table of n, a(n) for n=0..18.
Index entries for linear recurrences with constant coefficients, signature (31,-310,1240,-1984,1024).
FORMULA
a(n) = (2^n - 1)^4 - 2.
G.f.: (1984*x^4-2120*x^3+510*x^2-61*x+2) / ((x-1)*(2*x-1)*(4*x-1)*(8*x-1)*(16*x-1)). - Colin Barker, Apr 30 2013
EXAMPLE
a(0) = (2^0 - 1)^4 - 2 = 0^4 - 2 = -2.
a(1) = (2^1 - 1)^4 - 2 = 1^4 - 2 = -1.
a(2) = (2^2 - 1)^4 - 2 = 3^4 - 2 = 79.
MATHEMATICA
(2^Range[0, 20] - 1)^4 - 2 (* Paolo Xausa, Apr 19 2024 *)
PROG
(PARI) a(n)=(2^n-1)^4-2 \\ Charles R Greathouse IV, Feb 19 2016
CROSSREFS
Cf. A091516, A091515, A098878, A091514.
Cf. A093112, A098878.
KEYWORD
easy,sign
AUTHOR
Jonathan Vos Post, May 03 2006
EXTENSIONS
Offset changed to 0 by Paolo Xausa, Apr 19 2024
STATUS
approved
page 1

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